LETTERS TO THE EDITOR.
Euclid, Newton, and Einstein.
Since the results of the Eclipse Expedition of May last have been made public a very great deal of general interest has been displayed in a theory which, until a few weeks ago, was known only to mathematicians and physicists. Even among these, not many could offer any adequate explanation of the new view of space and time and their mutual relations, while some regarded the whole question as a mathematical joke which led to interesting results of no practical value; and probably not a few thought that a non-Euclidean system of geometry was inadmissible in any physical theory of the universe. On the other hand, there are some who have gone so far as to advocate that non-Euclidean geometry should be taught to boys and girls in secondary schools. The published books on this subject do not come into touch with any ordinary experience, and the whole subject, consequently, has been regarded as a mathematical fiction. So far from this being so, most people have actually seen the ordinary operations of life proceeding in non-Euclidean space, though they have not realised the meaning of all they have seen. In the space behind a plane mirror objects are reversed right and left (perverted), though in all other respects they correspond precisely to the real objects in front of the mirror of which they are the images, but in the space behind a convex mirror this is not the case. The geometry of this space and the behaviour of moving bodies therein, as viewed by the external observer and as studied by an intelligent being within the image space, say, the image of the external observer, who applies to the images and their movements the same standards of measurement as the external observer applies to the real objects in his own space, introduce us to a non-Euclidean space which is the subject of common observation, and prepare the mind for the reception of manv of the conclusions of the now famous theory of relativity. In the discussion of that theory two observers are supposed to be moving relatively to one another, each with his own set of measuring instruments and each living in his own world or system, and the differences between the phenomena which occur in each system as measured by the dweller in that system and by the external observer form the basis of the theory. Corresponding to these two observers we propose to consider the actual observer outside the convex mirror and his supposed intelligent image behind the mirror, and to consider how the images behind the mirror, treated as real objects, appear to behave to both observers.
In the first place, it is necessary to consider the size and shape of the objects, or, in other words, the geometry of the space. To save repetition it will be convenient to call the external observer . and his intelligent image B. The line joining the middle point of the mirror with the centre of the sphere of which the surface of the mirror is a part is the axis of the mirror, and may be supposed to be extended indefinitely outside the mirror. The image of an infinitely distant star on the axis of the mirror will be formed at a point half-way between the surface of the mirror and the centre of the sphere. This point is called the principal focus, and its distance from the mirror is the focal length, which is half the radius. It will be convenient to call this point F. A series of lines drawn from the circumference of the mirror outwards and all parallel to the axis encloses a cylindrical space to which the external objects considered are to be confined. All these lines produced indefinitely will at length meet the star on the axis of the mirror. Their images will, therefore, all converge to the principal focus F, and the whole of the infinite cylinder in the external world will correspond to a cone behind the mirror having F for its vertex and the mirror for its curved base. If an object outside moves away to infinity its image will never get beyond F, and the images of straight lines meeting the mirror and extending parallel to the axis as far as the distant star will all meet at F. We shall suppose the radius of curvature of the mirror to be very large as compared with the dimensions of the mirror itself or of the observer.
There is a very simple geometrical law connecting the distance of an object from the mirror and the distance of its image from F. This law need not concern us except to point out that as the object recedes from the mirror its image approaches F, and, as seen by the external observer, the dimensions of the image in all directions at right angles to the axis are proportional to its distance from F, but the dimensions parallel to the axis are proportional to the square of the distance from F of the image. This is the peculiar property of convex mirror space. If a cricket-ball is placed in front of the mirror at a distance equal to the focal length, its image will be half-way between the mirror and F, but the image will not be spherical. In all directions at right angles to the axis the dimensions will be reduced to one-half, but along the axis they will be reduced to one-quarter, so that the sphere will be represented by an oblate spheroid (an orange) with a polar axis one-half of the equatorial diameter. If the ball moves farther from the mirror the oblateness of the spheroid will be increased, and when the image is three-quarters of the way between the mirror and F the polar axis will be only one-quarter of the equatorial diameter of the spheroid, which will itself be only one-quarter of the diameter of the cricket-ball. If a circular hoop is placed with its plane at right angles to the axis its image will be circular, but if it is turned round so that its plane is parallel to the axis the image will be an ellipse, which will become more and more eccentric as the hoop recedes from the mirror and the image diminishes on approaching F. A top set spinning with its axis perpendicular to the axis of the mirror will appear in its image to the external observer to be elliptic, with its axes fixed in space, so that as any line of particles in the top approach parallelism to the axis of the mirror they will be squeezed together and expand again as they recede from parallelism. Midway between the mirror and F the density of the top will appear to A to be twice as great in the direction of the axis as in any direction at right angles to the axis, for the same number of particles will be squeezed into half the length.
All this has been written from the point of view of A, the external observer. But how will all these things appear to B, who is living and moving in the mirror space? Like A, the observer B may use a foot-rule for measuring length, breadth, and thickness, and a protractor for measuring angles. As A proceeds to measure the real object, B proceeds to measure the image, but as he approaches the focus his foot-rule, like himself and the image he is going to measure, gets smaller and in precisely the same proportion, so that if the image measured 6 in. in height when close to the mirror, it would always appear to measure 6 in. in height, for, as seen by A, the foot-rule would contract just as the image con-
